Min

Fig. 4.4 (a) Mutation-selection network of FAP initiation. We start with the type Xi because the first copy of the APC gene is inactivated in the germ line, (b) Mutation-selection network of HNPCC initiation. One mutation of an MSI gene is inherited, and therefore it takes only one step (inactivation of the second copy of the MSI gene, arrows up) to develop the MSI phenotype.

Again, the mutation-selection diagram can be converted into a system of ODEs. The solutions are given by

In the limit where wt/r —> oo, we have Z2{t) = nmu(u + Po)(t/r)2. If üt/r <C 1, then Z2(t) = nmNu(u + p0)(ü + p0)(t/T)3/3. X2(t) and Y2{t) are linear functions of time (there is one rate-limiting step), whereas Z2(t) grows slower than the second power of time (two rate-limiting steps plus one 'intermediate' step).

Some predictions of the model are shown in Table 4.5. The expected number of dysplastic crypts and the fraction of CIN crypts are calculated for t = 16 years. As the number of CIN genes, nc, increases, we expect more dysplastic crypts, and a larger fraction of crypts with CIN. According to our model, the expected number of dysplastic crypts grows linearly with time, and by the age of 16 years is expected to be in the thousands to tens of thousands, see Table 4.5. This should be compared with the observation that patients with FAP have hundreds to thousands of polyps by age 16.